This Wikipedia table (template for "Formal languages and grammars") maps grammar to language to abstract machine for more than a dozen languages.
But I miss another relation: Lambda calculus (that can be described syntactically as a certain context free grammar) has the same computational strength as a Turing machine. I somehow see this as functional programming (LC) vs imperative programming; but as I don't trust my intuition here, I want to explore the correspondence between abstract programming languages and automata further by asking this question.
Are there any similar equivalences between a calculus and an automaton, for finite state machines, and pushdown automata, to start with? (Other examples welcome.)